What is Phase space: Definition and 132 Discussions

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.

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  1. G

    Expressing phase space differential in terms of COM

    Homework Statement The Hamiltonian for a single diatomic molecule of identical atoms is given as $$H=\dfrac{\vec{p_1}\cdot\vec{p_1}}{2m}+\dfrac{\vec{p_2}\cdot\vec{p_2}}{2m}+\dfrac{K}{2}(\vec{r_1}-\vec{r_2})\cdot(\vec{r_1}-\vec{r_2})$$. Find the grand canonical partition function for a gas of...
  2. F

    Understanding The Meaning and Use of Phase Space

    Hi, I am trying to fully understand the meaning and usage of phase space in the various contexts it's used. For example particle physics, classical mechanics, statistical mechanics, thermodynamics, relativity. Also, there is configuration space, parameter space, and state space. How are all of...
  3. L

    Complex phase space coordinates

    First post ! I hope that my question will not make some long time physicists laugh. It is about geometrical quantization and the phase space in which we use : z=1/sqrt(2)(q+ip) My question is simple what is this 1/sqrt(2) ? And what is it is interpretation ? Thank you !
  4. S

    Volume in n Dimensions: Understanding the Meaning of n=0

    Hello, Surfing across the internet, I learned that the volume of a sphere in n dimensions can be expressed by V(n) = (Π^(n/2)) / Γ((n/2)+1), where n is the number of dimensions we are considering But if we consider n=0, then we get 1. So, how do we interpret this? I mean what does volume in zero...
  5. T

    Is the Dimension of Phase Space for a Rigid Diatomic Molecule Actually 12?

    Dimension of Phase Space of a rigid diatomic molecule is 10. Shouldn't it be 12?
  6. Urs Schreiber

    Insights Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively - Comments

    Urs Schreiber submitted a new PF Insights post Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively Continue reading the Original PF Insights Post.
  7. Jimster41

    Phase Space & SM: Universe Defined?

    Is the phase space of the universe, including it's partitioning, defined and calculated from the the SM?
  8. T

    I don't understand what is the meaning of phase? (phase space, phase point, etc)

    There is phase space, phase point, etc. So what is the exact meaning of phase? I only understand the definition of phase in wave. Beside what is canonical coordinate? What does this canonical mean?
  9. A

    Amount of microstates - phase space volume

    Homework Statement Dear all, I am desperately trying to solve the following exercise, but unfortunately can't find any resources how to properly calculate the phase space volume. Given is a system of ##N>>1## classical particles that are allowed to move in a cylinder with a Radius of ##R##...
  10. terra

    Volume elements of phase space

    First, two definitions: let ## \varrho (M)## be the probability density of macro states ##M ## (which correspond to a subgroup of the phase space) and ## \mathrm{d} \Gamma ## be the volume element of a phase space. In my lecture notes, the derivation for continuity equation of probability...
  11. T

    What makes phase space special?

    In Lagrangian/Hamiltonian mechanics, what is it that makes phase space special compared to configuration space? As a simple example, if I use ## q ## as my generalized position and ## v = \dot{q} ## as my generalized momentum, then the Hamiltonian H = \frac{1}{2} v^2 + \frac{1}{m} V(q) gives...
  12. Matta Tanning

    Relation between phase space and path integral formulation?

    I am trying to conceptually connect the two formulations of quantum mechanics. The phase space formulation deals with quasi-probability distributions on the phase space and the path integral formulation usually deals with a sum-over-paths in the configuration space. I see how they both lead...
  13. S

    Point in Phase Space and a Microstate

    Currently learning about Statistical Mechanics and just wanted to check my understanding. Am I right in saying that a point in phase space is just a specific microstate of the system?
  14. R

    Plotting Phase Space for Double Pendulum

    Homework Statement For a double pendulum, how do we plot the phase space for ##\theta_2## (the lower of the pendulum), i.e. the plot ##\theta_2, \ \dot{\theta}_2?## ##x## = horizontal position of pendulum mass ##y## = vertical position of pendulum mass ##\theta## = angle of pendulum (0 =...
  15. Y

    Phase space diagram for a spring in simple harmonic motion

    Homework Statement A mass m = 750 g is connected to a spring with spring constant k = 1.5 N/m. At t = 0 the mass is set into simple harmonic motion (no damping) with the initial conditions represented by the point P in the phase space diagram at the right. **(This phase space diagram has...
  16. naima

    Can the Displacement Operator Rotate a Photonic State in Phase Space?

    Hi PF I read the definition of the displacement operator: ##D(\lambda) = e^{\lambda a^\dagger - \lambda ^* a}## but i did not find how this operator can be implemented say in a cavity with a photonic state inside. Could you give me links? thanks.
  17. E

    Regression Analysis of Tidal Phases

    I have some 3-D model output for a river system that is tidally forced at the entrance. Right now, I'm trying to perform some linear regression on the harmonic constants of various tidal constituents at for several locations along the river compared to the observed tidal data. A linear...
  18. Quarlep

    How does Lagrange Mechanics work in coordinate space?

    I want to know lagrange mechanics work in phase space or in coordinate system.Leonard Susskind talked about the least action and he said If we know two point we can define trajectory but I don't know the diagram that he drow its a phase space or coordinate system (x,y,z,t) 19 min or...
  19. C

    Minimum volume of phase space

    In several books I have seen the statement that due to Heisenbergs principle no particle can be localized into a region of phase space smaller than ##(2 \pi \hbar)^3##. However, Heisenbergs uncertainty principle states that ##dx dp \geq \hbar/2## -- so a direct translation of this should imply...
  20. marcus

    Spin foam phase space, classical action (Wieland's talk is online)

    Wieland gave his ILQGS talk yesterday, 16 September. Audio and slides are on line. Title: Covariant loop quantum gravity: Its classical action, phase space and gauge symmetries http://relativity.phys.lsu.edu/ilqgs/wieland091614.pdf http://relativity.phys.lsu.edu/ilqgs/wieland091614.wav The...
  21. Logic Cloud

    Phase space in special relativity

    In classical mechanics we can get a nice overview of the dynamics of a system by looking at its position-momentum phase space. Is there a useful analogue of this concept in special relativity? Can the dynamics of a relativistic system be represented by its phase space in the same way as is done...
  22. L

    Physical pendulum in phase space

    Hi, I found out this paper http://www.pha.jhu.edu/~javalab/pendula/pendula.files/users/olegt/pendulum.pdf with this animation http://www.pha.jhu.edu/~javalab/pendula/pendula.files/users/olegt/pendula.html At first there is written there, that the area of possible states in some range...
  23. A

    How Does Phase Space Differ from Real Space in Dynamics?

    What exactly is phase space and how is it different from real space i.e. 3 co-ordinate system? what does it mean when someone says "dynamics occurs in phase space"? I'm very new to all this so pls take that into account to
  24. H

    Phase space and phase angles in a Quantum picture.

    Hi guys, I was hoping someone may be able to clear something up for me. I have been reading a paper on Quantum decoherence and was curious about what particular point (it could easily just be me misunderstanding) It is commonly noted that the superposition of waves represents all of the...
  25. sunrah

    Relationship between trace and phase space

    So I noticed we can define entropy in two very different ways: 1) quantum mechanically S = -k Tr(\hat{\rho}\ln{(\hat{\rho})}) 2) classically S = -k \int \rho \ln{(\rho)} d\Gamma where Tr is the trace and d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i} is the phase...
  26. Logic Cloud

    System and phase space trajectory

    To what extent do phase space trajectories describe a system? I often see classical systems being identified with (trajectories in) phase space, from which I get the impression these trajectories are supposed to completely specify a system. However, if you take for example the trajectory...
  27. L

    Phase space. Phase trajectories.

    If I understand well like in kinematics where we could have eq of motion ##x=x(t)##, ##y=y(t)## and we get eq of trajectory with elimination of time. In dynamics we have ##x=x(t)##, ##p=p(t)## and with elimination of time we get eq of phase trajectory. Am I right?
  28. S

    Exploring the Connection between Phase Space and Entropy in Particle Decay

    Is there any relationship between phase space and entropy? regarding to decay of particles
  29. J

    What Is the Dimensionality of Phase Space for a Two-Atom Molecule System?

    Homework Statement A classical gas consists of N molecules; each molecule is composed of two atoms connected by a spring. Identify the dimensionality of the phase space that can be used to describe a microstate of the system. The Attempt at a Solution I believe the answer is 12, but...
  30. A

    Wigner distribution in phase space

    This is about a specific property of the Wigner distribution in phase space. My professor mentioned the other day that the Wigner distribution treats all functions of momentum and space on the same footing as momentum itself or at least that's what I recall.He mentioned a specific problem where...
  31. S

    How to imagine a classical phase space for N particles?

    Classically a single particle will have 3 position coordinates and 3 momentum coordinates, and so it "exists" in a 6-dimensional phase space and moves around this space in relation to time (known as the phase trajectory). However I've read that when we have N classical particles, their...
  32. I

    Question about the uncertainty principle and unit cell in phase space

    In statistical mechanics, nearly all the textbooks say that the volume of the smallest cell in the phase space of a N-particle system is h^{rN} where h is the Planck Constant, r is the degree of freedom. Also these books say that this comes from the uncertainty principle. However, the...
  33. L

    How to define the Hamiltonian phase space for system?

    Title says it all, confused as to how I'm supposed to define the phase space of a system, in my lecture notes I have the phase space as {(q, p) ϵ ℝ2} for a 1 dimensional free particle but then for a harmonic oscillator its defined as {(q, p)}, why is the free particles phase space all squared...
  34. T

    Questions on phase space and CanonicalMicrocanonical ensembles

    1. Sketch the phase space of a weight free falling along the z coordinate (no motion in other directions). Sketch the trajectory of the free fall including impact on the ground. 2. Calculate the density of states, entropy, and temperature (all as a function of energy) for the following model...
  35. L

    Phase Space, Velocity, Integrals

    Hi, this is my first post. I did a search and in this sub-forum I found the most related threads for what I'm looking for. I need some guidance or where or how to learn all this mathematics for velocity-phase space integrals that appear in Maxwellian distributions. I'm an Engineer in...
  36. R

    Closed trajectories in phase space

    In general, how do you prove that a given trajectory in phase space is closed? For example, suppose the energy E of a one-dimensional system is given by E=\frac{1}{2}\dot{x}^2 +\frac{1}{2}x^2 + \frac{\epsilon}{4}x^4, where ε is a positive constant. Now, I can easily show that all phase...
  37. Spinnor

    2D H.O., graph phase space in R^2?

    Can I graph the phase space of a 2D harmonic oscillator in R^2 in the following way? Let one vector in R^2 represent for position of the point mass and let another vector represent momentum. Together these two vectors in R^2 can represent a single vector in R^4? Do we loose any "information" in...
  38. P

    Velocity phase space and energy diagrams

    Homework Statement Ok I have attached the pdf file and I have a problem with velocity phase spaces (Question 3a). Honestly, the lecture notes were not very helpful and looking online and in textbooks, they talked about solving Lagrange's equations but nothing to deal with the problem of Q3...
  39. L

    Classical phase space flow exact solution

    Homework Statement if i wanted to obtain an "exact" solution for flow s(t|k) k=(q,p) with a hamiltonian H(k) = x(ak) use the fact aJa = 0 where J is the poisson matrix Homework Equations The Attempt at a Solution I hate obscure proofs... i like actual question so I'm...
  40. H

    Coordinate System of Coupled Oscillators and 4D Phase Space representation

    Coordinate System of Coupled Oscillators and "4D" Phase Space representation So, I've modeled the interaction between two cantilever beams with the kinetic and potential energies shown in the above figure. The cantilevers are very stiff and have a small oscillation amplitude, so they can be...
  41. N

    Is entropy the volume in phase space of energy E or LESS than E?

    Hello, I thought the statistical definition of entropy for an isolated system of energy E (i.e. microcanonical ensemble) was S=k \ln \Omega where \Omega is the volume in phase space of all the microstates with energy E. However, if you take a look here...
  42. G

    Non intersecting phase space trajectories

    The phase space trajectories of an autonomous system of equations don't intersect. Can this be proved mathematically. Also what is the physical significance of this statement. What happens if they intersect?
  43. B

    Is (2, 3) a Closed Set in the Phase Space X = [0, 1] ∪ (2, 3)?

    X = [0, 1] \bigcup (2,3) is phase space. Show that (2, 3) open and closed set of X . the question is like that but I think it is false because it is not close, right?
  44. G

    Phase space trajectory question

    In my lecture they give the phase space picture for a simple pendulum http://mathematicalgarden.files.wordpress.com/2009/03/pendulum-portrait3.png?w=500&h=195 and then say that adjacent trajectories never diverge and therefore evolution is predictable. I wanted to ask, is the statement that...
  45. A

    How to go from the configuration space to the phase space?

    How can we go from the configuration space of the system to the phase space when velocity can be expressed in terms of momenta?
  46. BWV

    Hamiltonian Analysis in Curved Spacetime: Is it Possible in General Relativity?

    can you construct (or if yes, is it regularly done) a Hamiltonian in curved spacetime? If you took a system and moved it into a strong gravitational field or accelerated it to relativistic speeds can you still do Hamiltonian mechanics...
  47. marcus

    *Hidden Phase Space* at the Holo Cosmo corral (Verlinde video)

    Erik Verlinde gave this great talk at Perimeter on Wednesday last week, which is online as video. It was at a recent Holo Cosmo workshop they had there http://pirsa.org/C11010 It is a very exciting talk. He is actively grasping for what many people dream about: a concrete way to think of...
  48. M

    Why can't phase space trajectories intersect?

    Why can't trajectories in phase space intersect?
  49. S

    Phase space and the quantum Liouville theorem

    I would like to understand phase space better, spec. in relation to the quantum Liouville theorem. Can anyone point me to a decent online resource? I am most interested in conceptual understanding to begin with. Liouville's theorem says that if you follow a point in phase space, the number of...
  50. E

    Harmonic oscillator phase space integral

    Hi all, I am having trouble with a certain integral, which I got from Quantum Physics by Le Bellac: \int dxdp\;\delta\left( E - \frac{p^2}{2m} - \frac{1}{2}m\omega^2x^2 \right) f(E) The answer to this integral should be 2\pi / \omega\; f(E) . My attempts so far: This integral is basically a...
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